Computing device



Oct. 7, 1952 R. SERRLL. ETAL COMPUTING DEVICE 3 Sheets-Sheet l Filed May '7, 1949 Oct. 7, 1952 2,613,032

R. SERRELL ET AL COMPUTING DEVICE Filed May '7, 1949 3 Shee'LS-Sheet 2 ORNEY ATT Patented Oct. 7, 1952 UNITED r s'rirrirsy "PATENTTOFFICE r Princeton Junction, N. J., assignors to Radio Corporation of America, a corporation of Dela-` 'i Application May 7,1349, serial N6. 92,032

'1, l, This invention relates to computingdevices and more particularly to the provision of an improved fdevice and method of operation whereby the unknown quantities of linear simultaneous equa- 1tions, secular equations or polynomials may be determined. i j i l The device of the present invention is similar in many respects to the linear simultaneous equa- Itionmsolvers disclosed by` Brown and Goldberg `Patent No. 2,454,549 and `Brown Patent No. `2,455,974. i Y

l The present devicelike those of the aforesaid patents, utilizes Lthe amplitudes and relative `polarities` of a` plurality of voltages to represent the values and signs of the known quantities i theequatiOnS, and includes a plurality of ampliers interconnected by mutual feedback circuits in such a manner that the amplitude and relative polarity of the output voltage of each `ampliier represents the value and sign of a different unknown when there is` applied, (1) to Ythe input of each amplifier a voltage representative of the constant quantity of a diierent equation` and (2) to the feedback circuits of each amplifier voltages", res'presentative of `the coecients of the corresponding unknowns.

, The` present device is also like those of the patents in that the `adjustment required to establish the voltages representative of the known terms is effected by means of voltage dividers ofthe potentiometer type, each having an accurately calibrated dial scale adjustable to three significant places for determining the amplifier input voltages internas of a selectable percent of the reference voltage and for deriving from each amplifier output the desired fractional feedback voltages which represent the coefiicients of the unknowns.`

The present device is distinguished from those of the aforesaid patents in that it involves new .structural features and makes possible new methvods of computation. The principal physical modifications are (l) the use ofplural potentiometers for the matrix instead of single ones, (2) the addition of a number of amplifiers, `and (3) the addition oftwo or more groups of ganged potentiometers, rThe new methods of computation involve the provision of means necessary v.f or obtaining absolutely stable operation in all -cases and for adjusting the value of a complex determinant until it vanishes.

l Assuming the present device tohave a 12x12 matrix, it is operable to solve (1) systems of linear simultaneous equationsof all orders up to and including 12,l (2) secular equations of all forders up to and including 6 for real andcom- `plex roots (characteristic` roots of matrices or eigen values), and (3) polynomials of alldegrees up to and including 6 for real and] complex roots.

@Prineville Operating rangiofihedevi may' `comme. (ci. 2354-61) be increased by increasing `the size ofthe matrix and the number of its associated components.

"v i None` ofthe computing devices heretofore available appear to becap-able of solvingall these different types of equations with such simple equip` @ment as that herein disclosed. Ashereinafter shown, it4 includes 12 dials for setting up "the values of the known quantities Aand 144 dialsfor setting up the values of the coeflicients of the unknown quantities. In addition, two diaisjeach controllinga different group of ga'nged'` potenti--l v `ometers. are provided for the real and imaginary parts of the roots of secular equations and of polynomials. A switch and a dial for the'determination of` the unknowns (in a systemof linear simultaneous equations) anda cathode ray tube type of null indicator are `arranged in a manner similarto that disclosed'by the aforesaid patents. There is,l also provided a `switch Vfor `selecting the mode of operation of the device so that it will function to solvethe different types of equations. Meters may be provided'for simultaneously giving a continuous indication of the values of the unknowns in a system of linear equations. v i i The' principal objectof theinvention isfthe provision of an improved computing device, which is readily operated to solve different types of equations. Av further object lis the provision of a computing device` having such control elements thatitmay be readily adapted to determine` the unknown quantities of linear simultaneousequations,`secular equationsor polynomials. i The invention will be better understoodfrom thelfollowing description when considered in connection with the accompanying drawings. v v Referringto the drawings: i Fig. ,1; is a block diagram indicating the relation between the various parts of the improved computing device, i v v p Fig. 2 isa wiring ydiagram of the connections within one'voflthe boxes of Fig. 1,

Fig. 3` illustrates how certain of the potentiometers may be ganged together forsimultaneous adjustment. l

Fig. 4v illustrates a modied system of gauging such potentiometer which v, reduces the required number of potenticmeters, v

and thevbridge circuit indicatorare not shown in detail for the reason that they-are fully disclosed in the aforesaid patents and in an article by E. A.

Goldberg entitled Details of the simultaneous 'equation solver, RCA Review, vol. 9, No. 3,` Septemeer 194s, pp. 394-495; u l

ranged to be connected to the output of the generator I0 through a switch 23 and (2) are adjusted, as hereinafter explained, to provide/volt# ages which are representative of the' .c onstant quantities Y1, Y2 Yu and Yiz of theequations to be solved. The signalgenerator I0 pro.

duces an alternating reference voltage which may have a frequency of the order of 1000 cycles per second. The networks I I, I2, 2| and-22 are like the corresponding networks shown in lig.` 2

of Patent No. 2,454,549 within the dotted rectanglesYL-Yz Y. and identified by thereferrance-numeral II1.

The output` voltage of` each network II, |2-, .f 2| or 22 isappliedto the input of a different amplier24,`25, 34 or 35. The output voltageof the amplie1! 24 is applied through a f switch 36 and alead 31 to theinput of the networks 39, 39, 48 and 49 of the transposed matrix of thedevice. Similarly the output voltages of the'ampliers 25, 34 and 35 are applied respectively (l) to the networksll, 5I, 68 and 6|, (2) tothe networks 62,63, 12 and 13, and (3) to the networks 14', 15, 84 and 85 of the transposed matrix.

Output voltage is applied (l) fromthe networks 38, 50, 62 and 14 to the -input of an ampliner 86, (2) from the networks 39, 5|, 63, and 'I5 tothe input of anamplier81, (3) from the n-et- Worksv48, S0, 'I-2. and 84 tothe inputof anampliner 96, and (4) fromthenetworks 49',.6 13 and 85to the input of an amplifier 91.

The. output voltages` ofthe amplifiers. .86, 8.1, 96 and 91 arefedbacktothe networks of the main matrix of the device.I Thus feedback voltage is applied (1) from the amplifier 86 througha lead 9.8 tol thenetworks99, |00, |89. and vIII), (2) from theamplifier 81 through a-lead |41 to thenetworkslll, I|2, I2 and |22, (3) from the amplier 9 6 through a lead |48 to thenetworksl23, |24, |33 and |34,

and .(4) fromthe. amplier 91'through a lead |49 to the networksl35, |36, |45 and |46.

Output voltages are applied (1) from the networks 99,y III, |23 and|35 to the input ofthe amplier 24,*(2) fromthenetworks |89, ||2, |24 and |36` to theinput ofthe amplier25, (3) from the networks |09, |21, I33 and, |45 to theinput of the-amplier 3,4, and ,(4) from the networks IIB, |22, |34 and|46 tothe input ofthe ampliner 35.

The various networks of the main and transposed matrices of the device are similar. A detailed descriptionof the network 99 4ofthe main matrix therefore is sucientfor acompleteunderstanding of all the connections involved in the two matrices.

Fig. 2 shows the network 9.9 asincluding input terminals P and K which ,are connected respectively tothe anode and cathodeofthe amplier 86 from which this networkis energized Operating voltageis applied tothe amplifier from leads |50 and |,5I througha double-throw switch |52. In its upper closed .positionl lbteswitchL |52 connects (1) a xed resistor |53 in the anode lead oftheamplier 86 and (2)'apotentiometer |54 inthe ycathode lead ofthe amplifier 86. When the switch I52 .is in its lower closed position, the potentiometer |54,is in the anode lead and the resistor |53 is in the cathode lead. As explained in the aforesaid patents, this arrangementprovices for positive or negative values of the coeiiicients of. the unknown Ycluantitiespf the equations and insures that the'push-pull'output of theamplier is balanced at all times. Output voltage from the potentiometer |54 is applied through a summing resistor |55 to the input of the amplier 24.

Allfthe networks of both the main and transpose matrices contain a potentiometer, resistor and switch interconnected as described above. As so fardescribedl however, the system is substantially similar to the simultaneous equation solver previously referred to. One of the distinctions between the embodiment of the inventionbeing described herein and the above indicated usimultaneous equation solver is in the provision Aof an extra potentiometer, resistorand switch network similar tothe one just described whichisco-nnected'in parallel with thejnetwork' j ustdescribed. This is ,illustrated in'Figure 2, wherein the network 99 also includes a doublethrow switch |56 which is connected tothe output of the amplifier 86 in parallel with the switch |52; When the switch |56 is in its upper-closed position, aV xed resistor |51 is connectedin the anode lead and apotentiometer |56 is connected in the cathode lead. The connections ofthe resistor |51 and potentiometer |58 are reversed whenthe switch- |56'is in its lower closed position. Output voltage from the potentiometer |58'is applied through a'summing resistor |59 and a lead |69 to the input of the amplier 24.

The parallel potentiometer network shownin Figure '21s representative of the rectangles shown in Figure 1; which have two letters inscribed therein.Y In Figure 1 these are the networks along a principal diagonal of the-main matrix 99, IIE, |39-,A `|46-and the networksone row and one column adjacent Abothends of thev diagonal which include networks] 98, I I at one end the networks |34 and -I45'at the other end. Inthe transpose matrix the network Yalongthe principal diagonal having parallel potentiometer networks are networks 38,y 5I, 12 and 85, networks 39- and Y50 at one endl of fthe-diagonal and networks 13 andA 84 at the other end of the diagonal. As indicated above the identified rectangles-have twov letters inscribed therein,-

letters-a .and 'respectively represent the real and imaginary roots of a secular equation. One of the Aparallel'potentiometersin each network isused to -establish a voltage proportional to the known can coeiiicient. The other potentiometer is used .to determine the value of a or depending on which of the two functions it is assigned to represent. The proper polarity is determined by the switches. The algebraic addition indicated is provided by combining the outputs of the parallel potentiometers in the manner illustrated for the outputs of potentiometers |54 and |58 in Figure 2.

The fact that in Figure 1 double potentiometers lare indicated as being only along a, principal diagonal and at both ends of the diagonal is not to be taken asav limitation. The precise number of double potentiometer networks required for the solution of secular equations or complex roots of polynomial equations; is derived subsequently herein. All the'networks of the matrices may be of the double potentiometer type or only certain ones. Figure 1 is shown herein by way of illustrationand to aid in the explanation ofthe principles involved.

It-should be understood (1) thatall the 'networks of-the main and transposed matrices are 'the potentiometer tact member of Fig. 2. As indicated by similar. (2) that calibrated knobs simsalto those described in the aforesaid patentsmaybe provided for4 setting-,the contact `members |6| etc.

`to select voltages proportional tothe values oi the various coeicients of the unknown quantities of the equations, (3) that the polarity of the sei "lected voltage is determined'in each case by the Similarly the movable contact members of each 4,one of the parallel potentiometers assigned to represent the function by `which the imagin'aryvalue of a`root, is determined in a manner subsequently described `hereinfmay be all Aganged together asj` indicated `by the dash-dot line' |62 and operated from a single calibrated knob 66. `As indicated by the aforesaid patfents, the same isj true of the movable contact members which areoperated to set `corresponding coeflicientsintofthe two matrices.

`',Ihemovable contact' members of the u. potentiometers intercoupled as indicated by the broken line |62@ of Fig.` 1 may have their positions adjusted bya singlecalibrated knob |63 such as that shown by Fig. 3. In this ligure, each of the squares is intended to represent a potentiometer having its movable contact fixed to a shaft |64 which is.rotated by the knob |63. Similar means may be` provided for controlling the potentiometers which are ganged as indicated by the dash-dot line |6213 of Fig. 1.

Fig. 4 shows a modified control means by `which` a voltage representative of the algebraic sumoi the coefiicient am and root i` or may bederived from a single potentiometer such as |543with its movable con- Fig. 4, a clutch |65, |66, |61 or |63 isinterposed between4 the shafts to which the` respective movable contactM members t are xed. These -clutchesili may bekof anyfsuitable` type which iis Aelectrically or mechanically operated and (2)areldisengagedkwhile the valuesof. the coeiiicients an etc..ar`e. setinto the potentiometers 'by means ofthe calibrated knobs or dials |69,

|150,.|1|`, |`12,vetc. Thereafter, theclutches are @engaged androne of the knobs` maybe adjusted linlthe same'manner as knob |63 in .Figure 3 is adjusted todetermine the. value ofthe root a inthe manner subsequently described herein. 4This same type of control, of course,A is applicable toobviate the requirement forextra potentiometers andin other cases 'where the "output voltage ofa network is required to rep- `resent` the` algebraic sum of more than one `quantityof the equation.`

Where control means such as thatoflFig.. 4

ris used, no reversing switches, such as the "switchesi'lZ and |56 of Fig. 2] are required.. `In fstead there may.beprovded` a potentiometer l'control System suchas that. shown in Fig. 5. In this case, `there is providedan additional'amp ikfier |13Qwhich is .stabilized for gain asindicated bylthe feedback resistor |14 andisof the polarty` reversing type.' The output` voltage ofthe ampliiier 81, for example, is applied through a resistor |15 to the input of the amplifier |13. Theoutputvoltages of the amplifiers 81 and |13 are applied to opposite sections |16 and |11 of the potentiometer in such a way that the contact `member |18 may be operated to set into the device any desired plus or minus value of the coeiiicient a2-a The voltage so selected is applied through a summing resistor |19 to the input of the amplifier of Fig. 1 aspreviouslyindicated. l i

`When the device is utilized to solve linear simultaneous equations, the switches 23, 36 and |80 are in their illustrated closed positions and the values of the unknown quantities X1, X2, etc.

are indicated by tliemeters |6|, |82, etc. Alternatively these values may be measuredmore accurately by means of the bridge circuit described by the aforesaid patents. When the device is to be used for thesolution of secular equations, the switches 23, 36 and |80 are: lclosed at their S terminals and the .value of the unknown quantities is indicated by the reading shown by thecalibrations placed on the knobs which are used `to adjust the ganged a potentiometers and the ganged potentiometers. These a. and values are obtained after adjustment in a manner to be subsequently shown herein when an indicator (not shown) connected to the lead |85 indicates a zero voltage.

Method of solving linear simultaneous equations For stable operation, this method is like that described by Goldberg and Brown in an article entitled "An electrical simultaneous equation solver, Journal of Applied Physics, v01. `19, No. 4, April 1948, pp. 339-345. Also see Patent No. 2,455,974 to George W. Brown, issued December 14,1948.`

In considering the mathematical expressions which follow, it will be noted that the various subscripts of the cceilicients are not separated by a `hyphen as they are in Fig. 1` For example, the coefficient r11-1 applied to the network 99 of Fig. l appearsin the mathematical expressions as an. It is to be understood, however, `that the first digit or letter of the subscriptalways indicates the number of the equation and the fol'- lowing part of the subscript indicates its posi-tio in the equation. p

In accordance with the method set forth in `the Journal of Applied Physics, as well as Patent No. 2,455,974, the givensystem of equations of order n:

with matrix A, is replaced by an equivalent system with a matrix derived from A whose order is twice that of A, or 212,` and which may be written: i

,The Y `vector of the` original system jis'replaced by one which contains the n; originalfs.v and-n zero. As stated in the Journal of Applied Physics 9*- cally, ganged potentiometersior .,a'.\and another set foiz' the same .number-also mechanically ganged--for.

i To nd the` complex rootsof a secular equation,- the secular equation is ilrst expanded andl real terms collected and written in the forvmyof the equations of system (2). The matrix elements are then entered into :the machine exactly as shown in the (2n) `equations of` system `(2).

(Mgs) (m34-dic) :o

(Note` that for, n=6 this requires a. 1x10V for nfand (2)albori/febecomesH matrixfasprovided" inthe machine described l here). The fa and potentiometers are then rotated until theiirs't equation of (2) is satised, as shown by the nullindicator. r

Thus to solve the above secular equation lby l means of the devicelof Fig. 1, after reducing `the lthese conditions, the values indicated on the a and knobs or dials are the real and complex roots, respectively, of the secular equation. There will be as many roots (distinct or not) as equations. `The complex roots always occur in complex conjugate pairs.

The c ompam'ofz.` matriz method o! solving polynomials In accordance with this method, the device of Fig. 1 may be used to determine the real and complex roots of polynomials. The complex polynomial with real coeflicients of degree n may be written:

Now it is easy to see that, if we have )vda-Mn), the roots of the companion matrix:

*Pn-1 Pn-2 '-101 Po o 1 o o (5) are the same as those of the polynomial (4). The problem is thus reduced to that of finding the real and complex roots of a secular equation of the type of (1).

In the case of the polynomial equation (where r=ia1j and po to p11-1 are all real), by introducing n, (n=3), auxiliary complex vari- `This `determinant is setinto the equation solver as indicated by lthe legends applied to the networks of Fig. 6 which shows all thenetworksof the l2 12 matrices.

The procedure of solution is the same as that set forth above in connection with secular equations.

It follows that the complex roots of polynomials of all degrees up to and including 6 can be found by means ofthe device herein described exactly as constituted for the solution of equations previously discussed. The procedure is in most respects identical to that which has been explained above for the solution of secular equations, the only difference being that, in the case of polynomials, a number of matrix elements remain equal to zero. Y

What the invention provides is an improved device which is readily operated to solve different types of equations.

What is claimed is:

1. An equation solver including original and transposed matrices each comprising a plurality of interconnected elements each 0i" which elements includes a rst and second voltage divider of the potentiometer type, means for adjusting lthe polarity and value oi the output voltage of predetermined ones of said first voltage dividers according to the values of the known quantities of the equation to be solved, means for simultaneously adjusting the polarity and value of the output voltage of certain of the second of said voltage dividers according to the'values of the real roots of said equation, and means for simultaneously adjusting the polarity and value of the output voltage of other of said second dividers according to the values of the imaginary roots of said equation.

2. An equation solver including original and transposed matrices each comprising a plurality of interconnected elements each of which ele- `ments includes a rst and second voltage divider of the potentiometer type, means for adjusting the polarity and value of the output voltage of predetermined ones of said first voltage dividers 1 11 according to'the valuesof the knowny quantities of the equation to be solved, means for simultaneously adjusting thevpolarity and 'value of the output voltage of certain of the second of said voltage dividers according tothe vvalues of the real roots of said equation, means for simultaneously adjusting the polarity and value of the'output voltage'of otherof said second dividers according to the values .0f the imaginary roots of said equation, and means connected to one ofsaid matrices for indicating when the output voltages of said second voltage dviders'are such that said equation is satisfied.

3. A device asset forth in claim 2 wherein the means for adjusting the output voltages of the second voltage dividers include means for indicating the values of the real and' imaginary roots of thevequation.

4. An equation solver including original and transposed matrices each comprising a plurality of interconnected elements each of which elements includes a first and second voltage divider `of the potentiometer type, and means including of the;` potentiometer type, and '-rneans including" predetermined ones of said rst Voltage dividers according to the values of the known quantities of the equation to be solved, and means for adjusting the output voltages of said second voltage dividers to polarities and values such that the output of said matrices is zero.

ROBERT SERRELL. EDWIN A. GoLDnERe,V l

REFERNCES CITED The following references are` ofl record inthe" le of this patent:

An 'Automatici' Simultaneous Equation vlkCoinputer and itslUse in Solving Secular Equations, by W. A. Adcock;V the Review of Scientiiic 'Instruments-volume 19,'#3, March 1948. y -An "Electronic Simultaneous Squat.v Solver'g Journal of'Applied Physicsivolume 19, #4, pages 339-345, April 1948. 

